Rituximab and Bendamustine Hydrochloride, Rituximab and Ibrutinib, or Ibrutinib Alone in Treating Older Patients With Previously Untreated Chronic Lymphocytic Leukemia

Inclusion Criteria: * PRE-REGISTRATION (STEP 0) * All patients are REQUIRED to be pre-registered to A041202 in order to submit peripheral blood to the Alliance Hematologic Malignancy Biorepository (HEME) for central Zap-70 methylation. This specimen submission is mandatory prior to registration as results will be used for stratification * REGISTRATION (STEP 1) * Patients must be diagnosed with CLL in accordance with International Workshop on Chronic Lymphocytic Leukemia (IWCLL) 2008 criteria that includes all of the following: * \>= 5 x 10\^9 B lymphocytes (5000/uL) in the peripheral blood * On morphologic review, the leukemic cells must be small mature lymphocytes, and prolymphocytes must not exceed 55% of the blood lymphocytes * CLL cells on immunophenotype (performed locally) must reveal a clonal B-cell population, which express the B cell surface markers of CD19 and CD20, as well as the T-cell antigen CD5; patients with bright surface immunoglobulin expression or lack of CD23 expression in \> 10% of cells must lack t(11;14) translocation by interphase cytogenetics * Patients must be intermediate or high-risk Rai stage CLL * Intermediate risk (formerly Rai stage I/II) is defined by lymphocytosis plus enlarged lymph nodes at any site, with or without hepatomegaly or splenomegaly * High risk (formerly Rai stage III/IV) is defined by lymphocytosis with or without enlarged nodes and spleen plus disease-related anemia (hemoglobin \< 11 g/dL) or thrombocytopenia (platelet count \< 100 x 10\^9/L) that is not attributable to autoimmune hemolytic anemia or thrombocytopenia * Patients must meet criteria for treatment as defined by IWCLL 2008 guidelines which includes at least one of the following criteria: * Evidence of marrow failure as manifested by the development or worsening of anemia or thrombocytopenia (not attributable to autoimmune hemolytic anemia or thrombocytopenia) * Massive (\>= 6 cm below the costal margin), progressive or symptomatic splenomegaly * Massive nodes (\>= 10 cm) or progressive or symptomatic lymphadenopathy * Autoimmune anemia and/or thrombocytopenia that is poorly responsive to standard therapy * Constitutional symptoms, which include any of the following: * Unintentional weight loss of 10% or more within 6 months * Significant fatigue * Fevers \> 100.5 degrees F for 2 weeks or more without evidence of infection * Night sweats \> 1 month without evidence of infection * Prior treatment * Patients must not have had prior therapy for CLL (except palliative steroids or treatment of autoimmune complications of CLL with rituximab or steroids) * Treatment with rituximab and/or high dose corticosteroids for autoimmune complications of CLL must be complete at least 4 weeks prior to enrollment; palliative steroids must be at a dose not higher than 20 mg/day of prednisone or equivalent corticosteroid at the time of registration * Age \>= 65 years * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Patients with active hepatitis B defined by hepatitis B surface antigen positivity or core antibody positivity in the presence of hepatitis B DNA are not eligible for this study; patients with a positive hepatitis B core antibody but with negative hepatitis B DNA may participate, but must have hepatitis serologies and hepatitis B DNA monitored periodically by the treating physician * Intravenous immunoglobulin (IVIG) can cause a false positive hepatitis B serology; if patients receiving routine IVIG have core antibody or surface antigen positivity without evidence of active viremia (negative hepatitis B DNA) they may still participate in the study, but should have hepatitis serologies and hepatitis B DNA monitored periodically by the treating physician * Patients must not be receiving active systemic anticoagulation with heparin or warfarin; patients must be off warfarin therapy for at least 30 days prior to enrollment * Patients with class III or class IV heart failure by New York Heart Association, those with unstable angina, and those with uncontrolled arrhythmia are not eligible * Patients who have had a myocardial infarction, intracranial bleed, or stroke within the past 6 months are not eligible * Patients with known human immunodeficiency virus (HIV) are eligible if their CD4 count is \>= 350 cells/mm\^3 and if they are not taking prohibited CYP-interacting medications * Patients must not have any history of Richter's transformation or prolymphocytic leukemia (prolymphocytes in blood \> 55%) * Patients must not require more than 20 mg prednisone or equivalent corticosteroid daily * Patients must not have uncontrolled active systemic infection requiring intravenous antibiotics * Patients must not have continued requirement for therapy with a strong cytochrome P450 3A4/5 (CYP3A4/5) inhibitor or inducer * Patients must not have a known allergy to mannitol * Patients must not have prior significant hypersensitivity to rituximab (not including infusion reactions) * Patients may not have had major surgery within 10 days of enrollment, or minor surgery within 7 days of enrollment; examples of minor surgery include dental surgery, insertion of a venous access device, skin biopsy, or aspiration of a joint; the decision about whether a surgery is major or minor can be made at the discretion of the treating physician * Absolute neutrophil count (ANC) \>= 1,000/uL unless due to bone marrow involvement * Aspartate aminotransferase (AST) or alanine aminotransferase (AST) =\< 2.5 x upper limits of normal except if due to disease infiltration of the liver * Bilirubin =\< 1.5 x upper limits of normal (unless due to liver involvement, hemolysis, or Gilbert's disease) * Creatinine clearance \>= 40 mL/min * To be calculated by modified Cockcroft-Gault formula * Platelet count (untransfused) \>= 30,000/uL